On Harmonic Close-To-Convex Functions

In this paper, we study the family of sense-preserving complex-valued harmonic functions f that are normalized and close-to-convex on the open unit disk D. First we investigate the conditions for which f is close-to-convex on D. As a consequence, we derive a sufficient condition for f to be in this family. Using the condition, we establish sufficient conditions for f to be close-to-convex, in terms of the coefficients of the analytic and the co-analytic parts of f. Finally, we determine conditions on a, b such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)=zF(a,b;a+b;z)+\overline{\alpha z^{2}F(a,b;a+b;z)}$$\end{document} is harmonic close-to-convex (and hence univalent) in D, where F(a, b; c; z) denotes the Gaussian hypergeometric function. A similar result, and a number of interesting corollaries and examples of harmonic close-to-convex functions, are also obtained.